Some congruences involving binomial coefficients
| Autor: | Zhi-Wei Sun, Hui-Qin Cao |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | Colloquium Mathematicum. 139:127-136 |
| ISSN: | 1730-6302 0010-1354 |
| DOI: | 10.4064/cm139-1-8 |
| Popis: | Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant term in the expansion of $(1+x+x^{-1})^n$. We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k((-1)^k-(-3)^{-k})\equiv \left(\frac p3\right)(3^{p-1}-1)\ \pmod{p^3}.$$ In addition, we get some new combinatorial identities. 9 pages, final published version |
| Databáze: | OpenAIRE |
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